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Basic Definitions

A triangle is a polygon with three edges and three vertices.
Sum of interior angles: $A + B + C = 180^\circ$ (or $\pi$ radians).

Types of Triangles

Equilateral: All three sides are equal ($a=b=c$), all three angles are equal ($A=B=C=60^\circ$).
Isosceles: Two sides are equal ($a=b$), the angles opposite those sides are equal ($A=B$).
Scalene: All sides are different lengths, all angles are different.
Right-angled: One angle is $90^\circ$. The side opposite the right angle is the hypotenuse.
Acute: All three angles are less than $90^\circ$.
Obtuse: One angle is greater than $90^\circ$.

Area Formulas

Base and Height:
$Area = \frac{1}{2} \times base \times height = \frac{1}{2}bh$
Heron's Formula (given three sides $a, b, c$):
$s = \frac{a+b+c}{2}$ (semi-perimeter)

$Area = \sqrt{s(s-a)(s-b)(s-c)}$
Two Sides and Included Angle:
$Area = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B$
Coordinates of Vertices $ (x_1, y_1), (x_2, y_2), (x_3, y_3) $:
$Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

Perimeter

$P = a+b+c$

Right-Angled Triangle

Pythagorean Theorem: $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse).
Trigonometric Ratios:
- $\sin \theta = \frac{opposite}{hypotenuse}$
- $\cos \theta = \frac{adjacent}{hypotenuse}$
- $\tan \theta = \frac{opposite}{adjacent}$

Laws for General Triangles

Law of Sines:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ (where $R$ is the circumradius)
Law of Cosines:
- $c^2 = a^2 + b^2 - 2ab \cos C$
- $a^2 = b^2 + c^2 - 2bc \cos A$
- $b^2 = a^2 + c^2 - 2ac \cos B$

Special Points and Lines

Centroid: Intersection of medians (lines from vertex to midpoint of opposite side). Divides median in $2:1$ ratio.
Orthocenter: Intersection of altitudes (perpendiculars from vertex to opposite side).
Circumcenter: Intersection of perpendicular bisectors of sides. Center of the circumcircle.
Incenter: Intersection of angle bisectors. Center of the inscribed circle (incircle).
Euler Line: Connects the orthocenter, centroid, and circumcenter (in non-equilateral triangles).

Inradius and Circumradius

Inradius ($r$): Radius of the incircle.
$r = \frac{Area}{s}$
Circumradius ($R$): Radius of the circumcircle.
$R = \frac{abc}{4 \times Area}$

Medians

Length of median to side $a$ ($m_a$):
$m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}$

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